How randomized search heuristics find maximum cliques in planar graphs

Abstract

Diese Arbeit ist im Sonderforschungsbereich 531, "Computational Intelligence", der Universität Dortmund entstanden und wurde auf seine Veranlassung unter Verwendung der ihm von der Deutschen Forschungsgemeinschaft zur Verfügung gestellten Mittel gedruckt. ABSTRACT Surprisingly, general search heuristics often solve combinatorial problems quite sufficiently, although they do not outperform specialized algorithms. Here, the behavior of simple randomized optimizers on the maximum clique problem on planar graphs is investigated rigorously. The focus is on the worst-, average-, and semi-average-case behaviors. In semi-random planar graph models an adversary is allowed to modify moderately a random planar graph, where a graph is chosen uniformly at random among all planar graphs. With regard to the heuristics particular interest is given to the influences of the following four popular strategies to overcome local optima: local-vs. global-search, single-vs. multi-start, small vs. large population, and elitism vs. non-elitism selection. Finally, the black-box complexities of the planar graph models are analyzed. How Randomized Search Heuristics Find Maximum Cliques in Planar Graphs

Theoretical results in genetic programming: the next ten years?

“…Precise bounds on the expected run time for a variety of EAs have been obtained by using such techniques [116][117][118][119][120][121][122][123]. Typically, this has been done only for specific classes of functions although there are some exceptions where run-time bounds have been derived for more general combinatorial optimisation problems [124][125][126][127][128][129][130][131].…”

On the Maximum Number of Cliques in a Graph

“…Papadimitriou and Yannakakis [30] and Storch [44] proved that every n-vertex planar graph has O(n) cliques; see [7] for a more general result. The proof is based on the corollary of Euler's Formula that planar graphs are 5-degenerate.…”

Number of Cliques in Graphs with a Forbidden Subdivision

“…Our work can be viewed as an extension of Zykov's theorem [16] that establishes a bound on the number of cliques in graphs with no K t subgraphs. For planar graphs, Papadimitriou and Yannakakis [10] and Storch [12] proved a linear upper bound and finally Wood [15] determined the exact upper bound 8n − 16 for n-vertex planar graphs. Dujmović et al [3] generalized this result to graphs on surfaces.…”